It wasn't just the saddle that came off, it was the top portion of the seat post. Seat posts are usually press fit together between the main pipe and the top, and that's where yours has come apart. So there's a chance that, if you can remove the post intact from the frame, you could reassemble it. I had a seat post come apart once in a similar way, and I probably shouldn't be propagating this idea, but I "fixed" it by slathering JB Weld in the are where the two parts fit togehter. So far it's held up.
Best way to get any chemical persuasion in is to remove the bottom bracket, turn the bike upside down, and pour it in through the bottom of the seat tube. Then leave it upside down for several days. But first plug the top, either with tape or the like, or with the top portion of the seat post.
Final option, which may be worth considering since the seat post is now in two pieces - try one of the many destructive methods. RJ the Bike Guy (on youtube) recently posted a video showing a destructive method of removing the post by using an impact wrench on the pipe portion. Another option is to slice outward through the pipe with a hacksaw; just use caution as you get closer to the outside. RJ has several rather unorthodox options; just go to his channel and search on "stuck seat post".